| L(s) = 1 | + (−0.766 − 0.642i)2-s + (−0.766 − 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.552 + 3.13i)5-s + (0.173 + 0.984i)6-s + (−1.93 − 1.80i)7-s + (0.500 − 0.866i)8-s + (0.173 + 0.984i)9-s + (2.43 − 2.04i)10-s + (1.63 + 2.83i)11-s + (0.500 − 0.866i)12-s + (−0.412 − 2.33i)13-s + (0.325 + 2.62i)14-s + (2.43 − 2.04i)15-s + (−0.939 + 0.342i)16-s + (0.555 − 3.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−0.442 − 0.371i)3-s + (0.0868 + 0.492i)4-s + (−0.247 + 1.40i)5-s + (0.0708 + 0.402i)6-s + (−0.732 − 0.681i)7-s + (0.176 − 0.306i)8-s + (0.0578 + 0.328i)9-s + (0.770 − 0.646i)10-s + (0.492 + 0.853i)11-s + (0.144 − 0.249i)12-s + (−0.114 − 0.648i)13-s + (0.0869 + 0.701i)14-s + (0.629 − 0.528i)15-s + (−0.234 + 0.0855i)16-s + (0.134 − 0.763i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.000200464 + 0.0702232i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000200464 + 0.0702232i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (0.766 + 0.642i)T \) |
| 7 | \( 1 + (1.93 + 1.80i)T \) |
| 19 | \( 1 + (-1.05 - 4.22i)T \) |
| good | 5 | \( 1 + (0.552 - 3.13i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-1.63 - 2.83i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.412 + 2.33i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.555 + 3.14i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (0.941 + 0.342i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (5.94 + 2.16i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + 7.96T + 31T^{2} \) |
| 37 | \( 1 + (5.84 + 10.1i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.261 + 1.48i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (4.45 + 3.73i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.99 - 11.3i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (1.97 + 11.1i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-1.17 + 6.67i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (13.7 + 4.99i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (4.19 - 3.51i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-4.66 - 3.91i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (4.76 + 3.99i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (0.0768 - 0.0279i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-7.54 - 13.0i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (7.41 - 6.22i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (15.6 - 5.70i)T + (74.3 - 62.3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.944281002406542414637711899621, −9.372734929885553556275484492543, −7.78674346388233907834354902888, −7.28068870038237137924163209411, −6.70271227981243475739405033616, −5.59767843925252417383348673849, −3.95024988057363243096073967329, −3.20552534585989600922269352324, −1.91439094897023536464640692364, −0.04470260667164292126586550433,
1.49395838887641993187672550303, 3.44377558531401552413717155484, 4.62459452192502883709167848840, 5.51263749154414146160331807870, 6.20803997707403622400999487678, 7.24330802496641878842887013054, 8.537456890068372492249766578215, 8.956963353190466339756954263564, 9.492704835243690193277972370102, 10.57029745384057309999448162346
